For questions about linear approximations, $f(x) \approx f(a)+f'(a)(x-a)$ for $x$ around $a$.
Questions tagged [linear-approximation]
264 questions
3
votes
1 answer
How can I find closest line to $x^2$ in the $C^1[0,1]$ norm
Find all closest lines $p(x)=ax+b$ to $f(x)=x^2$ in the $C^1[0,1]$ norm. Note that the best approximation is not unique
Attempt : Let $r(x)=x^2-ax-b$.
Then $\|r(x)\|_{C^1}=\max\{|r^{(i)}(x)| : 0 \le i \le 1 \}$
$\|r(x)\|_\infty=r(0)=|b|$ or …
fivestar
- 919
2
votes
1 answer
Linear approximation of cos(28 degrees)
Evaluate cos(28 degrees) using linear approximation.
I have done this so far, but the answer seems to be incorrect and I can't figure out why.
y= 30
f(x) = cos(x)
f'(x) = -sin(x)
f(y) = f(30) = cos(30) = sqrt(3)/2
f(y) = f(30) = -sin(30) =…
Phantom
- 233
2
votes
2 answers
Why does one need to use $f(1,0)$ as linear approximation, rather than calculating $f(1.1, -0.1)$ directly?
Why does one need to use $f(1,0)$ a linear approximation, rather than calculating $f(1.1, -0.1)$ directly?
e.g. when $f(x,y)=xe^{xy}$ (used to be more complicated $e^{xy}+xye^{xy}$)
It's easy to see that $f$ is a not a linear function. however I…
mavavilj
- 7,270
2
votes
1 answer
When linearizing nth degree polynomials, is there any advantage in using Taylor series versus taking n derivatives?
If I need to get a linear approximation of a nonlinear function (linearize), for example approximate the values of a nonlinear function with a tangent line about point a, the two common choices seem to be derivation to obtain the slope of the…
milez
- 235
1
vote
1 answer
Most accurate linear approximation for two lines
Consider two lines defined by:
$$\begin{aligned}y_1 &= m_1 x + b_1\\y_2 &= m_2 x + b_2\end{aligned}$$
where for the sake of argument, the domain of both lines is the same and everything is a real number.
Is there an analytical solution to a new…
tpg2114
- 147
1
vote
0 answers
How to calculate the maximum absolute relative error using linear approximation?
Here is the problem
Using linear approximation, determine the maximum absolute relative error for the function:
$f(x,y,z) = \frac{−4⋅x^3⋅z}{y^3}$
at (1,3,2), assuming that the relative errors with respect to x, y and z are at most 0.8%, 0.3% and…
MatiasC
- 13
1
vote
1 answer
Linear approximation question where f(x) = g'(x) - how to use formula?
I'm a little confused by this question:
Lines and things that are linear are relatively boring in mathematics. What
if my function f(x) = g'(x). I’m going to ask the same question in a
different way.
What is an approximate value of…
MattE
- 113
- 4
1
vote
1 answer
Differentials: to the nearest milimetre question
When it says the side length of a cube was measured to be 20mm to the nearest millimeter. In this case, can I regard the maximum error for the side to be 0.5 mm? to compute the absolute error of the volume (if it was measured to be $8000mm^3$)? I am…
user1917231
- 235
1
vote
0 answers
Linear approximation for an implicit function 2
This is a question related to my previous question but due to wrong formulated my question I would like to re-post and not only edit since it was solved for the first time after I put 200 bounty points but I saw that it was edited to be…
Melina
- 937
1
vote
0 answers
Multivariable linear approximation: how can I estimate the error?
I'm learning about linear approximations, where a function $f(x,y) \approx L(x,y)$ if both are evaluated at the same point $(a,b)$ but $L(x,y)$ becomes more and more error prone when moving away from $(a,b)$.
I also know that the tangent plane in…
Floella
- 473
1
vote
0 answers
The best linear approximation
Good afternoon. I'm interesting in finding the best (or maybe good) linear approximation of the function $F^{(19)}: V_{56}\to V_8$, where
$$
\begin{array}{l}
F^{(1)}(x_1,x_2,\ldots, x_7) = P(x_2+x_6); \ F^{(2)}(x_1,\ldots,x_7)…
Mikhail Goltvanitsa
- 1,354
- 8
- 18
0
votes
2 answers
Proving that pressure is approximately equal to $P_0 \left(1 + \frac{k}{2} M \right)$
After conducting a series of experiments, a physicist concluded that the pressure around an object placed in a moving fluid is given by $$P(M) = P_0 \left( 1 + \frac{k - 1}{2}M \right)^{k/(k-1)},$$where $M$ is the square of the ratio of the speed of…
questionasker
- 1,148
0
votes
1 answer
linear approximation for ratio $\frac{(1+x)^n}{(1+y)^m}$
I have seen people use the approximation $\frac{(1+x)^n}{(1+y)^m}\approx1+nx-my$ where x and y are close to 0. I know $(1+x)^n \approx 1+nx$, but not sure about the ratio to difference approximation. Can someone please give an derivation?
user34829
- 235
0
votes
1 answer
how to show $\log\left(\frac{1+e^{a+bx}}{1+e^{a+c+bx}}\right)$ is approximately linear in x?
$X$ is random variable with the domain $(x_0,x_1)$. Under what conditions, can the function:
$$\log\left(\frac{1+e^{a+bx}}{1+e^{a+c+bx}}\right)$$ be approximately linear in $x$ (i.e., $k_0+k_1 x$)? $a, b, c$ are constant parameters.
Thanks,
Vincent
- 139
- 5
0
votes
1 answer
linear approximation method x=0.1+ln10
I was wondering how you would approach this question:
Estimate $e^x$
at $x = ln(10) + 0.1$, using the method of small increments (i.e. the linearisation
method).
Im not sure what to do i made $f(x) = e^x$ and $f'(x) = e^x$ but I'm not sure what i…
user639649
- 395